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Sample Question 9:
Step 1: Note that if the point \((2,3)\) is in the set \(S\), then by symmetry along the line \(y=x\), \((3,2)\) is also in \(S\).
Step 2: By symmetry about the origin, the points \((-2,-3)\) and \((-3,-2)\) are also in \(S\).
Step 3: By symmetry about the x-axis, the points \((2,-3)\) and \((3,-2)\) are in \(S\).
Step 4: By symmetry about the y-axis, the points \((-2,3)\) and \((-3,2)\) are in \(S\).
Step 5: Thus, every point of the form \((\pm 2, \pm 3)\) or \((\pm 3, \pm 2)\) must be in \(S\), and all these 8 points satisfy all of the symmetry conditions. Therefore, the smallest number of points in \(S\) is 8 (Option D).
A set \(S\) of points in the \(xy\)-plane is symmetric about the origin, both coordinate axes, and the line \(y=x\). If \((2,3)\) is in \(S\), what is the smallest number of points in \(S\)?
\(\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16\)
Answer Keys
Question 9: DSolutions
Question 9Step 1: Note that if the point \((2,3)\) is in the set \(S\), then by symmetry along the line \(y=x\), \((3,2)\) is also in \(S\).
Step 2: By symmetry about the origin, the points \((-2,-3)\) and \((-3,-2)\) are also in \(S\).
Step 3: By symmetry about the x-axis, the points \((2,-3)\) and \((3,-2)\) are in \(S\).
Step 4: By symmetry about the y-axis, the points \((-2,3)\) and \((-3,2)\) are in \(S\).
Step 5: Thus, every point of the form \((\pm 2, \pm 3)\) or \((\pm 3, \pm 2)\) must be in \(S\), and all these 8 points satisfy all of the symmetry conditions. Therefore, the smallest number of points in \(S\) is 8 (Option D).